Problem on Rotational mechanics / Conservation of angular momentum.

In summary, the question asks for the angular speeds of two cylinders after they stop slipping due to friction. The conversation discusses two methods of solving the problem: using the conservation of angular momentum and equating the angular impulse due to friction to the change in angular momentums of the cylinders. The final answer is given for both methods.
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Homework Statement


The figure shows two cylinders rotating about their axes, all the variables involved are shown.
The two cylinders are moved closer to touch each other keeping the axes parallel. The cylinders first slip over each other at the point of contact but slipping finally ceases due to friction between them.

Find the angular speeds of the cylinders after the slipping ceases.
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Homework Equations



  1. After slipping ceases, if w1' and w2' are new angular speeds then...
    w1' * r1 = w2' * r2
  2. Angular momentum = Iw.

The Attempt at a Solution


I have succeeding in solving the problem by equating the angular impulse due to friction to the change in angular momentums of both cylinders and eliminating friction force from the two equations.

But I'd like to solve this using conservation of angular momentum as well since it should be pretty straightforward and easy.

Since no external torque acts on the system, conserving angular momentum, I get:

[tex]I_{1}\omega_{1}+I_{2}\omega_{2}=I_{1}\omega_{1}' + I_{2}\omega_{2}'[/tex]

Using this with w1' * r1 = w2' * r2, I get

[tex]\omega_{1}' = \frac{I_{1}\omega_{1}+I_{2}\omega_{2}}{I_{1}r_{2}+I_{2}r_{1}} r_{2}[/tex]
and a similar expression for w2'.

The answer, however, is [tex]\omega_{1}' = \frac{I_{1}\omega_{1}r_{2}+I_{2}\omega_{2}r_{1}}{I_{1}r_{2}^{2}+I_{2}r_{1}^{2}} r_{2}[/tex]
 
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Related to Problem on Rotational mechanics / Conservation of angular momentum.

1. What is rotational mechanics?

Rotational mechanics is a branch of classical mechanics that deals with the motion of objects that rotate around a fixed axis. It involves the study of concepts such as angular velocity, torque, and moment of inertia.

2. What is the conservation of angular momentum?

The conservation of angular momentum is a fundamental principle in physics that states that the total angular momentum of a closed system remains constant. This means that the angular momentum of a system cannot be created or destroyed, only transferred between different objects within the system.

3. How is angular momentum calculated?

Angular momentum is calculated by multiplying an object's moment of inertia by its angular velocity. The moment of inertia is a measure of an object's resistance to rotational motion, and the angular velocity is the rate of change of an object's angular position.

4. What is the relationship between torque and angular acceleration?

Torque and angular acceleration are directly proportional to each other, according to Newton's Second Law for rotational motion. This means that a larger torque will cause a greater angular acceleration, and vice versa.

5. How is the conservation of angular momentum applied in real-world situations?

The conservation of angular momentum is applied in many real-world situations, such as the motion of planets in the solar system, the movement of spinning tops, and the flight of objects like frisbees and boomerangs. It is also used in engineering and design to optimize the performance and stability of rotating machinery and vehicles.

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